Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T07:50:20.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Criteria for components of a function space to be homotopy equivalent

Published online by Cambridge University Press:  01 July 2008

GREGORY LUPTON  and
SAMUEL BRUCE SMITH
GREGORY LUPTON
Affiliation:
Department of Mathematics, Cleveland State University, Cleveland OH 44115, U.S.A. e-mail: [email protected]
SAMUEL BRUCE SMITH
Affiliation:
Department of Mathematics, Saint Joseph's University, Philadelphia, PA 19131, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y) have the same homotopy type. We describe certain group-like actions on map(X, Y). Our basic results assert that if maps f, g: X → Y are in the same orbit under such an action, then the components of map(X, Y) that contain f and g have the same homotopy type.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 1 , July 2008 , pp. 95 - 106
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Allaud, G.. On the classification of fiber spaces. Math. Z. 92 (1966), 110–125.CrossRefGoogle Scholar
[2]Crabb, M. C. and Sutherland, W. A.. Counting homotopy types of gauge groups. Proc. London Math. Soc. (3) 81, no. 3 (2000), 747–768.CrossRefGoogle Scholar
[3]Dold, A.. Partitions of unity in the theory of fibrations. Ann. of Math. (2) 78 (1963), 223–255.CrossRefGoogle Scholar
[4]Dold, A.. Halbexakte homotopiefunktoren. Lecture Notes in Mathematics, vol. 12 (Springer-Verlag, 1966).CrossRefGoogle Scholar
[5]Dugundji, J.. Topology (Allyn and Bacon Inc., 1978), Reprinting of the 1966 original, Allyn and Bacon Series in Advanced Mathematics.Google Scholar
[6]Fuchs, M.. Verallgemeinerte homotopie-homomorphismen und klassifizierende Räume. Math. Ann. 161 (1965), 197–230.CrossRefGoogle Scholar
[7]Golasinski, M. and Mukai, J.. Gottlieb groups of spheres, preprint.Google Scholar
[8]Gottlieb, D. H.. On fibre spaces and the evaluation map. Ann. of Math. (2) 87 (1968), 42–55.CrossRefGoogle Scholar
[9]Gottlieb, D. H.. Evaluation subgroups of homotopy groups. Amer. J. Math. 91 (1969), 729–756.CrossRefGoogle Scholar
[10]Hansen, V. L.. Equivalence of evaluation fibrations. Invent. Math. 23 (1974), 163–171.CrossRefGoogle Scholar
[11]Hansen, V. L.. The homotopy problem for the components in the space of maps on the n-sphere. Quart. J. Math. Oxford Ser. (2) 25 (1974), 313–321.CrossRefGoogle Scholar
[12]Hansen, V. L.. On spaces of maps of n-manifolds into the n-sphere. Trans. Amer. Math. Soc. 265 no. 1, (1981), 273–281.Google Scholar
[13]James, I. M.. General Topology and Homotopy Theory (Springer-Verlag, 1984).CrossRefGoogle Scholar
[14]Mimura, M., Lee, K.-Y. and Woo, M. H.. Gottlieb groups of homogeneous spaces, Topology Appl. 145, no. 1–3, (2004), 147–155.Google Scholar
[15]Kono, A.. A note on the homotopy type of certain gauge groups. Proc. Roy. Soc. Edinburgh Sect. A 117, no. 3–4, (1991), 295–297.CrossRefGoogle Scholar
[16]Kono, A. and Tsukuda, S.. A remark on the homotopy type of certain gauge groups. J. Math. Kyoto Univ. 36, no. 1, (1996), 115–121.Google Scholar
[17]Kono, A. and Tsukuda, S.. 4-manifolds X over BSU(2) and the corresponding homotopy types Map(X, BSU(2)). J. Pure Appl. Algebra 151, no. 3, (2000), 227–237.CrossRefGoogle Scholar
[18]McClendon, J. F.. On evaluation fibrations. Houston J. Math. 7, no. 3, (1981), 379–388.Google Scholar
[19]Milnor, J.. On spaces having the homotopy type of a CW complex. Trans. Amer. Math. Soc. 90 (1959), 272–280.Google Scholar
[20]Møller, J. M.. On spaces of maps between complex projective spaces. Proc. Amer. Math. Soc. 91, no. 3, (1984), 471–476.CrossRefGoogle Scholar
[21]Mislin, G., Hilton, P. and Roitberg, J.. Localization of Nilpotent Groups and Spaces (North-Holland Publishing Co., 1975), North-Holland Mathematics Studies, No. 15, Notas de Matemática, No. 55. [Notes on Mathematics, No. 55.]Google Scholar
[22]Siegel, J.. G-spaces, H-spaces and W-spaces. Pacific J. Math. 31 (1969), 209–214.CrossRefGoogle Scholar
[23]Stasheff, J.. A classification theorem for fibre spaces. Topology 2 (1963), 239–246.CrossRefGoogle Scholar
[24]Sutherland, W. A.. Path-components of function spaces. Quart. J. Math. Oxford Ser. (2) 34, no. 134, (1983), 223–233.CrossRefGoogle Scholar
[25]Sutherland, W. A.. Function spaces related to gauge groups. Proc. Roy. Soc. Edinburgh Sect. A 121, no. 1–2, (1992), 185–190.CrossRefGoogle Scholar
[26]Tsukuda, S.. Comparing the homotopy types of the components of Map(S4,BSU(2)). J. Pure Appl. Algebra 161, no. 1–2, (2001), 235–243.CrossRefGoogle Scholar
[27]Varadarajan, K.. Generalised Gottlieb groups. J. Indian Math. Soc. 33 (1969), 141–164.Google Scholar
[28]Whitehead, G. W.. On products in homotopy groups. Ann. of Math (2) 47 (1946), 460–475.CrossRefGoogle Scholar
[29]Whitehead, G. W.. Elements of homotopy theory. Graduate Texts in Mathematics, vol. 61 (Springer-Verlag, 1978).CrossRefGoogle Scholar
[30]Yoon, Y. S.. On n-cyclic maps. J. Korean Math. Soc. 26, no. 1, (1989), 17–25.Google Scholar